"After reviewing ..., it ...": is it grammatical?

I was interested in the sentence “After reviewing the special commissioner's report, it is obvious to me that Mr. Namnum abused his responsibility and privileges to secure a job for his wife, who was clearly unqualified,” which appeared in news article, titled “School Administrator Resigns After Inquiry Into Wife's Job,” on The New York Times (March 28, 2012).

The schools chancellor, Dennis M. Walcott, issued a statement saying: "After reviewing the special commissioner's report, it is obvious to me that Mr. Namnum abused his responsibility and privileges to secure a job for his wife, who was clearly unqualified. No one at any level of government should be allowed to put his own self-interest ahead of his obligation to the city."

For a non-native English speaker, it seems more natural rewording the mentioned sentence in this way: "A review of the special commissioner's report provides evidence that Mr. Namnum abused ...".

Are there rules governing this strange grammatical structure (After reviewing ..., it ...)?

It's not uncommon, but OP's example is not grammatically correct - it's a dangling participle. Here's Grammar Girl's take on it, if you want to read more.

Grammatically speaking, the first word following the introductory clause “After reviewing the special commissioner's report" should be a noun (the subject, who did the reviewing). A valid rewording would be, for example, "After reviewing the...report, I find it obvious that...".

Some people may think it's pedantic to criticise such usage, but I think most careful writers (and even speakers) would avoid it.

It's grammatically correct. The word "After" indicates a subordinating conjunction.

In your example the clause

After reviewing the special commissioner's report

is the dependent clause. The dependent clause can't exist on it's on (it doesn't really make sense) and it's purpose is to add cause or background to the independent clause. But it does need an independent clause.

The independent clause:

it is obvious to me that Mr. Namnum abused his responsibility and privileges to secure a job for his wife, who was clearly unqualified.

... is a normal sentence which could be on its own. It just lacks the context like in this case the additional information on why it's obvious to the writer, that Mr. Mamnun abused his responsibility.

The independent clause doesn't require any special pronouns (like the it). For example, this sentence is grammatically valid:

After reviewing the special commissioner's report, my head started to hurt

It's perfectly grammatical, but there is a long history of people who think there is something wrong with this quite normal construction, so if you don't want to fall foul of the self-appointed grammar police, you'd best avoid it.